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32
Markov chain monte carlo convergence diagnostics
 JASA
, 1996
"... A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise ..."
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Cited by 367 (6 self)
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A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler convergence, including applying diagnostic procedures to a small number of parallel chains, monitoring autocorrelations and crosscorrelations, and modifying parameterizations or sampling algorithms appropriately. We emphasize, however, that it is not possible to say with certainty that a finite sample from an MCMC algorithm is representative of an underlying stationary distribution. 1
Geometric ergodicity of Metropolis algorithms
 STOCHASTIC PROCESSES AND THEIR APPLICATIONS
, 1998
"... In this paper we derive conditions for geometric ergodicity of the random walkbased Metropolis algorithm on R k . We show that at least exponentially light tails of the target density is a necessity. This extends the onedimensional result of (Mengersen and Tweedie, 1996). For subexponential targe ..."
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Cited by 85 (2 self)
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In this paper we derive conditions for geometric ergodicity of the random walkbased Metropolis algorithm on R k . We show that at least exponentially light tails of the target density is a necessity. This extends the onedimensional result of (Mengersen and Tweedie, 1996). For subexponential target densities we characterize the geometrically ergodic algorithms and we derive a practical sufficient condition which is stable under addition and multiplication. This condition is especially satisfied for the class of densities considered in (Roberts and Tweedie, 1996).
Convergence of slice sampler Markov chains
, 1998
"... In this paper, we analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastic monotone, and deduce analytic bounds on the total varia ..."
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Cited by 65 (9 self)
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In this paper, we analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastic monotone, and deduce analytic bounds on the total variation distance from stationarity of the method using FosterLyapunov drift condition methodology.
Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems
 Statistical Science
"... This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain method ..."
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Cited by 49 (5 self)
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This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...
MCMC methods for continuoustime financial econometrics

, 2003
"... This chapter develops Markov Chain Monte Carlo (MCMC) methods for Bayesian inference in continuoustime asset pricing models. The Bayesian solution to the inference problem is the distribution of parameters and latent variables conditional on observed data, and MCMC methods provide a tool for explor ..."
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Cited by 41 (1 self)
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This chapter develops Markov Chain Monte Carlo (MCMC) methods for Bayesian inference in continuoustime asset pricing models. The Bayesian solution to the inference problem is the distribution of parameters and latent variables conditional on observed data, and MCMC methods provide a tool for exploring these highdimensional, complex distributions. We first provide a description of the foundations and mechanics of MCMC algorithms. This includes a discussion of the CliffordHammersley theorem, the Gibbs sampler, the MetropolisHastings algorithm, and theoretical convergence properties of MCMC algorithms. We next provide a tutorial on building MCMC algorithms for a range of continuoustime asset pricing models. We include detailed examples for equity price models, option pricing models, term structure models, and regimeswitching models. Finally, we discuss the issue of sequential Bayesian inference, both for parameters and state variables.
MCMC Methods for Financial Econometrics
 Handbook of Financial Econometrics
, 2002
"... This chapter discusses Markov Chain Monte Carlo (MCMC) based methods for es timating continuoustime asset pricing models. We describe the Bayesian approach to empirical asset pricing, the mechanics of MCMC algorithms and the strong theoretical underpinnings of MCMC algorithms. We provide a tuto ..."
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Cited by 34 (4 self)
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This chapter discusses Markov Chain Monte Carlo (MCMC) based methods for es timating continuoustime asset pricing models. We describe the Bayesian approach to empirical asset pricing, the mechanics of MCMC algorithms and the strong theoretical underpinnings of MCMC algorithms. We provide a tutorial on building MCMC algo rithms and show how to estimate equity price models with factors such as stochastic expected returns, stochastic volatility and jumps, multifactor term structure models with stochastic volatility, timevarying central tenclancy or jumps and regime switching models.
Estimation and Inference via Bayesian Simulation: An Introduction to Markov Chain Monte Carlo
, 2000
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Gibbs Sampling
 Journal of the American Statistical Association
, 1995
"... 8> R f(`)d`. To marginalize, say for ` i ; requires h(` i ) = R f(`)d` (i) = R f(`)d` where ` (i) denotes all components of ` save ` i : To obtain Eg(` i ) requires similar integration; to obtain the marginal distribution of say g(`) or its expectation requires similar integration. When p i ..."
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Cited by 28 (0 self)
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8> R f(`)d`. To marginalize, say for ` i ; requires h(` i ) = R f(`)d` (i) = R f(`)d` where ` (i) denotes all components of ` save ` i : To obtain Eg(` i ) requires similar integration; to obtain the marginal distribution of say g(`) or its expectation requires similar integration. When p is large (as it will be in the applications we envision) such integration is analytically infeasible (the socalled curse of dimensionality*). Gibbs sampling provides a Monte Carlo approach for carrying out such integrations. In what sorts of settings would we have need to mar
Estimating Ratios of Normalizing Constants for Densities With Different Dimensions
 Statistica Sinica
, 1997
"... Abstract: In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the cu ..."
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Cited by 17 (4 self)
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Abstract: In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the current Monte Carlo methods such as the bridge sampling method (Meng and Wong (1996)), the path sampling method (Gelman and Meng (1994)), and the ratio importance sampling method (Chen and Shao (1997)) cannot directly be applied. In this article, we extend importance sampling, bridge sampling, and ratio importance sampling to problems of different dimensions. Then we find global optimal importance sampling, bridge sampling, and ratio importance sampling in the sense of minimizing asymptotic relative meansquare errors of estimators. Implementation algorithms, which can asymptotically achieve the optimal simulation errors, are developed and two illustrative examples are also provided.
ON THE COMPUTATIONAL COMPLEXITY OF MCMCBASED ESTIMATORS IN LARGE SAMPLES
"... In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasiBayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the LaplaceBernsteinVon Mises central limit theorem, which states that ..."
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Cited by 14 (3 self)
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In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasiBayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the LaplaceBernsteinVon Mises central limit theorem, which states that in large samples the posterior or quasiposterior approaches a normal density. Using this observation, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases, where the underlying loglikelihood or extremum criterion function is possibly nonconcave, discontinuous, and of increasing dimension. However, the central limit theorem restricts the deviations from continuity and logconcavity of the loglikelihood or extremum criterion function in a very specific manner. Under minimal assumptions for the central limit theorem framework to hold, we show that the Metropolis algorithm is theoretically